Studies on the analytic behaviour of number theoretic L-functions

(1)引入了广义多重Zeta函数的概念，它是Euler-Zagier多重和函数和Barnes多重Zeta函数的推广，并利用Mellin-Barnes积分公式证明了它们的解析延拓和渐近展开式。作为应用，我们证明了Hurwitz Zeta函数的高阶矩的渐近展开式，以及高维球面的拉普拉斯函数行列式的显式公式。(2)找到了连接Ramanujan公式、模关系和近似函数方程的基本原理，并证明了与多个Zeta函数有关的各种L函数的级数表达式的快速收敛。(3)引入普适性理论中的正密度方法，证明了SL(2，II)或其同余群的尖点形式上的自同构L函数和Rankin-Selberg L函数的普适性，以及Lerch Zeta函数的联合普适性。

(1) We introduced the notion of generalized multiple zeta-functions, which is a generalization of both the Euler-Zagier multiple sums and the Barnes multiple zeta-functions, and, by using the Mellin-Barnes integral formula, proved their analytic continuation and asymptotic expansions. As applications, we proved asymptotic expansions of higher power moments of Hurwitz zeta-functions, and also explicit formulas of determinants of the Laplacians of high-dimensional spheres.(2) We found a basic principle connecting Ramanujan's formula, modular relations, and approximate functional equations, and proved rapidly converget series expressions of various L-functions, in connection with multiple zeta-functions.(3) We introduced the positive density method in universality theory, and proved the universality of automorphic L-functions, and Rankin-Selberg L-fanctions, attached to cusp forms of SL (2, II) or its congruence sabgroups, also the joint universality of Lerch zeta-functions.